Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)
Chapter 2: Semi-Classical LightMatter Interaction
2.1 A Two-level System Interacting with Classical Electromagnetic
Field in the Absence of Decoherence
2.1.1 Hamiltonian for Interaction between Light and a Two-level System
Consider a two-level system, say an electron in a potential well or in an atom with two energy levels,
interacting with electromagnetic radiation of frequency  . The electric field at the location of the twolevel system
 is,
E t   nˆEo cost   
2
e2
e1
Eo cos( t +  )
1
In the absence of the electric field the Hamiltonian of the electron is,
Hˆ o   1 e1 e 2   2 e1 e 2
The energy difference between the upper and the lower state is    2   1 . In the presence of the
electric field, the potential energy of the electron of charge q  e  is,
 

 q rˆ. E t    qrˆ.nˆE cost   
o
So the Hamiltonian becomes,

Hˆ t   Hˆ o  qrˆ.nˆ Eo cost   
Note that the Hamiltonian is time-dependent. In the two dimensional Hilbert space consisting of only


states e1 and e2 , and assuming e1 rˆ e1  e2 rˆ e2  0 , the above Hamiltonian can be written as,
Hˆ t    1 e1 e1   2 e2 e2

  R cost    e1 e2
 cost    e2 e1

where the frequency R is related to the “dipole moment” of the states,


 R  qEo e2 r .nˆ e1  qEo e1 r .nˆ e2
In the so called rotating wave approximation only the important resonant term in each cosine term is
retained (we will discuss this in detail later in the course) and one obtains,
 R
Hˆ t    1 e1 e1   2 e2 e2 
expi t  i  e1 e2  exp i t  i  e2 e1
2
 R
expi t  i  ˆ   exp i t  i  ˆ  
  1 Nˆ 1   2 Nˆ 2 
2
The Hamiltonian above is used to describe the interaction of a classical electromagnetic field with a twolevel system. Any of the methods used for the time-independent two-level system problem can be used to
solve the time-dependent problem as well.


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Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)
2.1.2 Solution Using the Schrodinger Picture
We assume a time-dependent solution of the form,

i 1t
e 
 t   c1 t 
e1  c 2 t 
and plug it into the Schrodinger equation,
d  t 
 Hˆ  t 
i

i 2 t
e 
e2
dt
to get the following equations for the coefficients,

 i t  i
d c1t 

 i R c 2 t  e 
dt
2

i t  i

d c 2 t 
 i R c1t  e 
2
dt
The detuning  is defined as    2   1    . The above equations can be solved using appropriate
2
   2 . Suppose,
boundary conditions. It is convenient to define a frequency  as    R
 t  0  e1 , then the solution is,
c1t   e
i

t
2
i

t  i
2

    
 
cos 2 t   i  sin 2 t 





 R
  
i  sin 2 t 



In the absence of detuning (i.e.   0 ) , the probabilities of finding the electron in the upper and lower
levels oscillate with a frequency equal to  R . These oscillations are called Rabi oscillations, and the
frequency  R is called the Rabi frequency. The maximum value of the population difference,
c 2 t   e
c 2 t   c1 t  , is +1. In the presence of detuning, the populations oscillate with a frequency equal to
2
2
 and the maximum value of the population difference, c 2 t   c1 t  , is
2
2
2
R
   2
2
R
   2
 1. There is
no simple way to incoporate decoherence and/or population decay from the upper level into the lower
level in the Schrodinger equation. To include these we have to use the density operator formalism.
2.1.3 Solution by Transformation to a Time-Independent Hamiltonian
Consider the time-dependent Hamiltonian describing the interaction of a two-level system with light,
 R
Hˆ t    1 e1 e1   2 e2 e2 
expi t  i  e1 e2  exp i t  i  e2 e1
2
 R
expi t  i  ˆ   exp i t  i  ˆ  
  1 Nˆ 1   2 Nˆ 2 
2
We define a unitary operator B̂t  as follows,
Bˆ t   exp  i Nˆ t


1


To understand the effect of the unitary operator B̂t  on the state  t  suppose that  t  is written as
follows,
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Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)
 t   c1 t 

i 1t
e 
e1  c 2 t 

i 2 t
e 
e2
then,
Bˆ t   t   c1t  e
  
i 1
t


i 2 t
e 
e1  c 2 t 
e2
The operator B̂t  “boosts” the energy of the lower level by  . It is easy to prove the following two
identities,
Bˆ t   exp  i Nˆ 1 t  1  Nˆ 1 exp i t   1


where,


Bˆ t   Nˆ 1  Hˆ t  Bˆ  t   Hˆ R

 R i
Hˆ R   1    Nˆ 1   2 Nˆ 2 
e ˆ   e  i ˆ 
2
The Schrodinger equation for the state  t  is,

  t 
 Hˆ t   t 
t
Suppose we define a new R t  state as follows,
 t   Bˆ t   t 
i
R
Then differentiating both sides with respect to time, and using the above identities, gives,
 R t 
 Hˆ R R t 
i
t
The above equation shows that the original problem with a time-dependent Hamiltonian Ĥ t  is
equivalent to a problem with a time-independent Hamiltonian Ĥ R . One can solve the time-independent
problem easily using any of the methods of the last Chapter. In particular, some insight is obtained by
using the eigenstates of the Hamiltonian Ĥ R . Suppose, for simplicity, that the detuning
   2   1    is zero. Then the two eigenstates, and the corresponding eigenenergies, of the
Hamiltonian Ĥ R are,
v1 

e
2
1

e
2
1
i
2
e1  e  i  2 e2


1   2 
 R
2
 R
2
The slitting of the eigenenergies is  R . Suppose the initial state is,
v2 
i
2
e1  e  i  2 e2
 t  0  R t  0   e1 
1   2 
e i  2
2
 v1
 v 2 .
Then,
R (t ) 
 R (t )
i
 Hˆ R t

e
i
 i

( 2   R 2)t
e  i  2   ( 2   R 2)t
R (t  0)  R (t ) 
e
v1  e 
v2 

2 


i
 2t
e 
i
 2t  i
 
 
cos R t  e1  ie 
sin R t  e2
 2 
 2 
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Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)
Finally, the desired quantum state  t  can be obtained as follows,
i
i
 2 t  i
 
 
 t 
cos R t  e1  ie 
sin R t  e2
 2 
 2 
The above result agrees with the one obtained in 2.1.2 when detuning is zero. The most interesting aspect
of the above technique is that it shows that the time-dependent problem can be mapped onto a timeindependent problem. We know that in any time-independent problem the eigenstates play a special role.
Eigenstates are stationary states and whenever any arbitrary state is written as a superposition of
stationary states, the probability for the state to be found in any one of the stationary states upon
measurement remains time-independent. Suppose,  t  0  R t  0   v1 . Then,
 Bˆ  t  R t   e 
 1t
R (t ) 
i
( 2   R 2)t
e
  t 
 Bˆ  t  R t   Bˆ  t  e 
v1
i

e
i
R
t
2
2
( 2   R 2)t
v1

i
 i  t i 

 t i
e  1 2 e  e  2 2 e 
1
2




And therefore,
1
2
2
2
1
e2  (t )  e2  (t  0)

2
The probabilities of finding the electron in the upper and lower levels remain time-independent when the
initial state corresponds to an eigenstate of the time-independent Hamiltonian Ĥ R . This conclusion
remains valid, of course, even when detuning is non-zero.
e1  (t )
2

e1  (t  0)
2

2.2 Optical Bloch Equations
2.2.1 Solution Using the Density Operator and Optical Bloch Equations
Here we will use the density operator approach in the Schrodinger picture. Starting from the density
operator equation,
d ˆ t 
i
 Hˆ t , ˆ t   Hˆ t  ˆ t   ˆ t  Hˆ t 
dt
one can derive the differential equations for 11 t  ,  22 t  ,  21t  , and 12 t  by taking matrix


elements of the density operator equation with respect to the states e1 and e2 . The result is,

d 11t 
 i R  21t  expi t  i   12 t  exp i t  i 
dt
2

d  22 t 
 i R  21t  expi t  i   12 t  exp i t  i 
dt
2

 2  1

d 12 t 
i
12 t   i R expi t  i   22 t   11t 

dt
2

 2  1
R
d  21t 
 i
 21t   i
exp i t  i   22 t   11t 

dt
2
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Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)
The detuning  between the field frequency and the energy level separation is,
   2   1   
The solution is most easily obtained, and insightful as well, if one uses a formalism originally developed
to treat precession of nuclear spins. We define three new real quantities V x t  , Vy t  , and Vz t  as
follows,

Vx t    21t  e i t  i  12 t  e  i t  i


Vy t   i  21t  e i t  i  12 t  e  i t  i

Vz t    22 t   11t 

The vector V t  is,

V t   V x t  xˆ  V y t  yˆ  Vz t  zˆ

Note that the x- and y-components of the vector V t  are related to the off-diagonal elements of the

density operator, and the z-component of the vector V t  is related to the occupation probability
difference between the upper and the lower energy levels (also called the population difference). One can
now rewrite the equations for the density matrix elements in terms of the quantities V x t  , V y t  , and
Vz t  ,
d V x t 

  V y t 
dt

d V y t  
 V x t    R Vz t 
dt

d Vz t 
  R V y t 
dt



If one defines a vector frequency  as    R xˆ  zˆ , then the equations above can be written in

vector formvery compactly as,

d V t  
  X V t 
dt
Note that all fast time dependencies (on the scale of  or  2   1   ) present in the equations for the

density matrix elements are absent in the equation for the vector V t  and this is the main advantage of



working with V t  . The equation for V t  resembles the equation of the magnetic moment M t  of a

classical spin
in
a
constant
magnetic
field
B
,



d M t 
  B X M t 
dt
where  is the gyromagnetic ratio (ratio between the spin magnetic moment and the spin angular
momentum). The
 magnetic moment equation was derived by Felix Bloch in 1946. Therefore, the equation
for the vector V t  is sometimes called the optical Bloch equation.

The following facts are not hard to prove and follow directly from the vector equation for V t  ,

i)
The magnitude of the vector V t  does not change with time.
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Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)
ii)
iii)

The vector V t  executes a periodic motion and the angular frequency is equal to the

magnitude of the vector  .

The “plane of rotation” is the plane in which the tip of the vector V t  lies during rotation.

The vector  is always normal to the plane of rotation.
Plane of rotation (the front facet of the cone)

Cone
Circle of rotation (solid line)
2.2.1 The Bloch
Sphere

The vector V t  completely characterizes the quantum state of an electron interacting with an
electromagnetic wave.
y-axis
Bloch Sphere
V(t
)
x-axis
z-axis
The z-component of the vector equals Vz t    22 t   11 t  and gives the population difference

between the two-levels and can have values between +1 and -1. The other components of the vector V t 
capture the coherence in the system and are related to the off-diagonal components of the density
operator. Since the magnitude of the vector V t  does not change with time, and if at time t  0 the


quantum state of the system is e1 then V t  0   Vz t  0 zˆ   zˆ , the magnitude of the vector V t  is
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Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)

unity. Therefore, V t  rotates with time but its tip always lies on a sphere of radius unity which is called
the Bloch sphere, as shown in the Figure.
Case of Zero Detuning: If the initial quantum state of the electron is e1 , then

V t  0   Vz t  0 zˆ   zˆ , and the vector tip at time t  0 is at the location indicated by the black dot
on the sphere in the Figure. If electromagnetic field with zero detuning (i.e.   0) is turned on at t  0 ,


then since    R xˆ , the tip of the vector V t  rotates in the y-z plane (with x-axis as the axis of
rotation)
as shown in the Figure. The y-z plane is the plane of rotation in this case. The tip of the vector

V t  follows a circle formed where the plane of rotation intersects with the Bloch sphere. This circle is

called the circle of rotation. At any time, the projection of the vector V t  onto the z-axis (i.e. the zcomponent) gives the population difference  22 t   11 t  . The projection onto x-axis and y-axis
gives the coherences (i.e. the off-diagonal components of the density matrix). At time when

 t  R t   2 , the tip reaches the south pole, and the population difference goes to zero but the
coherence is maximum. At this stage, the electron wavefunction (obtained by direct computation using
the Schrodinger equation) is,
1



 i 2 t  i
1  i  t

 t  
e
e1  i e
e2 

2

 t 
2 R

When  t   R t   , the tip reaches the positive z-axis, the electron is in the upper level, and the
population difference is maximum (+1) but the coherence is minimum. The electron wavefunction at this
stage (obtained by direct computation using the Schrodinger equation) is,
 t   i

 i 2 t  i
e 
e2 t  
R


When  t   R t  3  2 , the population difference goes to zero again. And when  t   R t  2 , the
population
 difference is at its minimum value of -1, the electron is in the lower state, and the tip of the
vector V t  is at the starting location shown by the black dot in the Figure. Thus, in one Rabi period

(given by 2   2  R ) the population difference goes through one complete cycle. This
phenomenon is called Rabi flopping or Rabi oscillation.
Case of Non-Zero Detuning: Using the Bloch sphere picture, one can visually solve many problems.


Suppose there is some detuning so that the axis of rotation, given by the direction of    R xˆ  zˆ ,


is slightly tilted away from the negative x-axis, as shown below in the Figure. The vector V t  starts from
the location of the black dot, but now the plane of rotation is not the y-z plane but a plane that contains

the black dot (since this is the initial starting point) and is also perpendicular to the direction given by  .
The best way to see what this means is to look at the sphere from the top (i.e. from above the North
 pole)
as shown in the Figure below. As the detuning is increased, the plane of rotation of the vector V t  tilts.

The tip of the vector V t  , of course, always remains on the Bloch sphere and in the plane of rotation. As


the detuning increases, the frequency of rotation (given by  ) increases and the tip of the vector V t 
7
Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)
rotates in circles of smaller and smaller radii (the circle of rotation formed by the intersection of the plane
of rotation
 and the Bloch sphere). The maximum population difference, given by the z-component of the
vector V t  , decreases from +1, and at large detunings it is no longer even positive at any time during the
dynamics.
Plane of rotation when =0
(the y-z plane)
z-axis
Plane of rotation when
 is small



x-axis
Plane of rotation when
 is large
Bloch sphere seen from
above the “North pole”
2.3 A Two-level System Interacting with Classical Electromagnetic
Field in the Presence of Decoherence and Population Decay
2.3.1 Optical Bloch Equations with Decoherence and Population Decay
We know that electrons in materials or in atoms in higher lying energy levels usually come down to lower
energy levels by giving off their energy to phonons or to any other non-radiative channel. In a two-level
system, one can include relaxation of the population from the upper level to the lower level via such nonradiative mechanisms. We assume that the population relaxation time is T1 . One can also include the
effects of decoherence by assuming that the off-diagonal components of the density matrix decay with a
time constant T2 . In physical systems, the same scattering mechanism is often, but not always, the source
of both population relaxation (i.e. T1 ) and decoherence (i.e. T2 ). When this is true, T1 and T2 are
related; T2  2T1 . Otherwise, T2  2T1 . In the presence of population relaxation and decoherence, the
equations for the elements of the density matrix become,

d 11t   22 t 

 i R  21t  expi t  i   12 t  exp i t  i 
dt
T1
2
 t  
d  22 t 
  22  i R  21t  expi t  i   12 t  exp i t  i 
dt
T1
2
8
Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)
   1   t   i R expi t  i   t    t 
 t 
d 12 t 
  12  i 2
12
22
11

dt
T2
2
 t     1 

d  21t 
   12  i 2
 21t   i R exp i t  i   22 t   11t 

dt
T2
2
The equations for the components, V x t  , V y t  , and Vz t  are now as follows,
d V x t 
1

  V x t   V y t 
dt
T2

d Vy t 
1

  Vy t   V x t    R Vz t 

dt
T2
V t   1   V t 
d Vz t 
 z
R y
dt
T1
One can nolonger write the above set of equations in the compact form,

d V t  
  X V t 
dt

Also note that the magnitude of the vector V t  is not conserved anymore. The above equations have a
well defined steady state solution,
   2 
 1   T2  
  
 
V t    
z
2
V y t    
V x t    
 
2
1   T2    R
T2T1


  RT2
2
 
2
1   T2    R
T2T1


 
 T2   RT2


2
 
2
1   T2    R
T2T1


The above expressions for the steady state show that electromagnetic radiation, no matter how strong,
cannot create population inversion in steady state (i.e. make the z-component Vz t  positive in steady
state).
Optical Control by Short Pulses: Short pulses of light can be used to control and/or prepare desired
quantum states of a two-level system. For example, consider a situation where detuning is zero and
population relaxation and decoherence times are sufficiently long. Suppose,  t  0  e1 . Radiation

is switched on at time t  0 and switched off at time given by  t   R t   . Such a radiation pulse is
called a  pulse. At the end of the pulse, the value of Vz t  is approximately +1 and population
inversion is achieved. The quantum state of the two-level system at the end of the  pulse is,
 t   i

 i 2 t  i
e 
e2 t  
R
9
Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)
The population will, of course, relax into the lower state after the pulse because of T1 . Similarly, as

shown earlier, a  2 pulse (i.e. a pulse whose duration in time is  2    2R ) can be used to take
an electron from an initial state e1 into a linear superposition of e1 and e2 given by,
 t 
1



 i 2 t  i
1  i  t

e
e1  i e
e2 


2
 t 

2 R
2.4 Photon Echo Experiments
Photon echo is a useful experimental technique to characterize decoherence times in materials. Consider
an isolated two-level system (in the absence of any radiation) in which the mean position of the electron
in both states, e1 and e2 , is zero, and,



d  e2 r e1  e1 r e2
Now consider the superposition state,

  i 1 t

i 2 t


 i


e1  e
e
e2 
i  e
  t   e


2




The mean position of the electron in such a superposition state is,


 

 t  r  t   d cos 2 1 t   



The charge density associated with the electron wavefunction oscillates in space with a frequency related
to the energy level difference. Electron in such a superposition state can therefore radiate electromagnetic
energy, just like a classical dipole antenna. Radiation from one such electron is too difficult to detect.
Radiation from many such electrons in a collection of two-level systems would also be difficult to detect
unless all the electrons were oscillating in-phase. The trick is to get many electrons to oscillate in-phase.
Electron states in many materials, such as semiconductors, can be modeled as a collection of two-level
systems. In most cases, electron energies of the upper and/or levels would have a range of values. In
other words, the energy difference   , or the detuning   measured with respect to some fixed  ,
would not be the same for all the two-level systems. For some systems,  would be zero, for some 
would be positive, and for some  would be negative. Therefore, charge oscillations in different twolevel systems would not stay in-phase for long. Note that the superposition state,

  i 1 t

i 2 t

 i

 e 
 e1  e  i e2 

e
e
e
e
1
2 
   t   e i 
 t  0   e i 




2
2







corresponds to a vector V t  with the following components,
 e  e  i e2 

 t  0   e i  1


2






Vx t   cos t     
Vy t   sin t      Vz t   0




The argument of the cosine and the sine is also the time-dependent
phase of the charge oscillation relative

to  t . If for two different two-level systems the vectors V t  were to become identical at any point in
time, then at that moment the electron charge in these two two-level systems will be oscillating in-phase.
10
Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)
Photon echo techniques are able to achieve in-phase charge oscillations in a collection of two-level
systems with even different values of   . Consider such a collection of two-level systems interacting
with radiation of frequency  . Suppose a short and strong (  R    ) electromagnetic  2 pulse is
used to excite the two-level systems assumed to be all initially in the ground state. The dynamics of each
two-level system are governed by the equations,
d V x t 
1

  V x t   V y t 
dt
T2

d Vy t 

1
  Vy t   V x t    R Vz t 

dt
T2
V t   1   V t 
d Vz t 
 z
R y
dt
T1
Assume
that decoherence and population relaxation times are very long. Right after the pulse, the vector

V t  equals  ŷ for all the two-level systems, as shown in Fig.(a) below. The subsequent evolution of
the two-level systems is according to the equations,
d Vy t 
d V x t 
d Vz t 


  V x t 
  V y t 
0
dt

dt
dt


The above equations show that the vectors V t  of two-level systems with different detunings move apart
in the x-y plane as shown below in Fig.(b). After some time, say Td , another short and strong (

 R    ) electromagnetic  pulse is used. The  pulse rotates the vectors V t  of all two level

systems by 180-degrees around the x-axis and the final positions of the vectors V t  are as shown in
Fig.(c) below. Right after the  pulse, the free evolution of the two-level systems is again according to
the equations,
d Vy t 
d V x t 
d Vz t 


  V x t 
  V y t 
0
dt

dt
dt

The  pulse changed the sign of the V y component of every two-level system, but did not change the

sign of the V x component. Consequently, the vectors V t  of two-level systems with different detunings,
which were moving apart in the x-y plane before the  pulse, now start moving closer with time. At time

equal to exactly Td after the  pulse, the vectors V t  of al two-level systems come together, as shown
in Fig.(d). At this point in time, electrons in all the two-level systems are oscillating in-phase.
Consequently, the radiation emitted by them can easily be detected. Because of different detunings,
charge oscillations in different two-level systems will soon go out of phase with each other. In actual
experiments, a weak but detectable radiation pulse is detected at the moment when the electrons in all the
two-level systems are oscillating in-phase. This pulse is called the photon echo pulse.
Now suppose decoherence is present. The coherences will get reduced the longer the time delay Td is
compared to the decoherence time T2 and, therefore, the strength of the photon echo pulse will also get
reduced as Td is made longer. Thus, if the strength of the photon echo pulse is measured as a function of
the time delay Td then this information can be used to extract the decoherence time T2 (assuming
T1  T2 ).
11
Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)
y
y
(a)
(b)
x
x
V
V
z
z
y
y
(c)
(d)
V
V
x
z
x
z
2.5 Ramsey Fringes and Atomic Clocks
Consider the problem of determining the frequency  of a radiation source with high accuracy (fractional
accuracy better than one part in 1014). One way to do this would be to make an atom, or a two-level
system, interact with the radiation from the source. Suppose the energy level separation of the two-level
system is known to be  with a very high accuracy. The frequency of the source is assumed to be close
to   but the detuning   is unknown and needs to be determined with high precision. Suppose, one
considers the following scheme: the two-level system is made to interact with the radiation from the
source for a duration Tint . Immediately afterwards, the upper state occupation (i.e.  22 ) is measured.
The question is whether this absorption experiment can be used to determine   . We will assume that
decoherence and population relaxation times are very long. Suppose the duration Tint is chosen such that


2
   2 and
the radiation appears to the two-level system as a  pulse (  Tint   ). But    R
12
Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)
  is unkown. To overcome this problem, one can deploy a sufficiently strong pulse such that
R    . After the pulse, the value of  22 is,
 22 
2
R
2
R
   2
If R    , the above expression shows that  22 is close to unity irrespective of the value of   .
Therefore, this simple absorption experiment will not be able to determine   with high precision.
Atomic coherences can be exploited to obtain much better accuracies, as we will see now.
2.5.1 Ramsey Fringes
Consider a two level system prepared in the ground state, i.e.  t  0  e1 . At time t  0 , a strong
 2 pulse ( R    ) of duration Tint and detuning   excites the system. The dynamics of the
two-level system are governed by the equations,
d V x t 
1

  V x t   V y t 
dt
T2

d Vy t 

1
  Vy t   V x t    R Vz t 

dt
T2
V t   1   V t 
d Vz t 
 z
R y
dt
T1
Assume
that decoherence and population relaxation times are very long. After the pulse, the state vector

V t  equals  ŷ (Fig.(a) below). The subsequent free evolution of the two-level system is according to
the equations (Fig.(b) below),
d Vy t 
d V x t 
d Vz t 


  V x t 
  V y t 
0
dt

dt

dt
The vector V t  rotates in the x-y plane with a frequency equal to   for a duration T (Fig.(b) below).
After this duration T , a second  2 pulse of duration Tint interacts with the two-level system. If the


time duration T is such that   T  m , where m is any positive odd integer, and the state vector


V t  equals  ŷ (Fig,(c) below) , then the second  2 pulse will make the state vector V t  equal to
 ẑ and the two-level system will be in the lower state e1 after the second pulse. On the other hand, if


the time duration T is such that   T  m , where m is any positive even integer, and the state


vector V t  equals  ŷ (Fig,(d) below), then the second  2 pulse will make the state vector V t  equal
to  ẑ and the two-level system will be in the upper state e2 after the second pulse. It is not difficult to
show that under the assumptions R    and very long decoherence and relaxation times, the value
 
of Vz at the end of the second pulse is given by cos T  , and the occupancy of the upper state is given
 

  
by 0.5 1  cos T  . Therefore, if one makes a measurement on the two-level system after the second
  

pulse and determines whether the system is in the upper state or the lower state, then, knowing T , the
detuning   of the radiation from the energy level separation of the two-level system can be determined
13
Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)
to a very high accuracy. Note that frequency accuracy in this case is not determined by the time of
interaction Tint of the two-level system with the radiation but by the duration T between the two  2
pulses which can be made very long subject to the constrains imposed by decoherence and relaxation
times.
y
y
(a)
(b)
x
x
V
V
z
z
y
y
(c)
(d)
V
V
x
z
x
z
2.5.2 Cesium Atomic Clocks
The principle discussed above is used in atomic clocks which are very stable and high-precision
frequency sources. In a Cesium atomic clock, radiation from a tunable RF oscillator is coupled to Cesium
atoms which have an energy level separation close to ~9.192 GHz. The frequency of the RF oscillator can
drift over time and needs to be locked to a stable reference with high precision. The principle of operation
of the clock is as follows. Cesium atoms are prepared in the ground state and made to interact twice with
 2 pulses from the RF source. The duration between these pulses is T . At the end of the second pulse,
the upper state occupancy of the Cesium atoms is determined. This procedure is repeated many times
while the frequency of the RF source is varied. A typical plot of the measured upper state occupancy after
the second pulse vs the detuning of the RF source is shown in the Figure below.
14
Upper state occupancy
Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)
The observed oscillations are called Ramsey fringes (after Norman Ramsey). Note that the width of the
center fringe (in Hz) is 1 T (see the inset). A more detailed analysis shows that if one does not make the
assumption  R    and also does not assume that decoherence and relaxation times are infinitely
long, then the observed envelope of the oscillations can also be reproduced. Using feedback from these
measurements, the frequency of the RF oscillator is adjusted and locked to the energy level separation of
the Cesium atoms. In an actual Cesium fountain clock (see Figure above), Cesium atoms are prepared in
the ground state and hurled upwards against gravity (like water in a fountain). They pass through a
microwave a cavity twice; on their way up and then again on their way down. While passing through the
cavity, the atoms interact with the radiation from the RF source. Finally, the state of the Cesium atoms is
detected and the cycle is repeated. Recently, ultra-cold atoms with temperatures less than K have been
used to reduce decoherence and relaxation rates resulting in very stable atomic clocks. The root-meansquare fractional frequency stability obtained in typical Cesium atomic clocks is given by the expression,
Tc
1
1

~

T  Na
Here, Tc is the cycle time (time needed to complete one measurement),  is the integration time used in
estimating the frequency, and Na is the number of atoms used in one cycle. Typical values of Tc and
Na are 1 second and 6  10 5 atoms, respectively. This gives,

3  10 14
  is in seconds


Very stable RF sources, with fractional stabilities smaller than 10-16, have been realized using these
principles.
~
15